Temperature and Pressure Dependence of the Properties of the Liquid

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Temperature and Pressure Dependence of the Properties of the Liquid-Liquid Interface. A Computer Simulation and Identification of the Truly Interfacial Molecules Investigation of the Water-Benzene System Lı´via B. Pa´rtay,† George Horvai,‡,§ and Pa´l Jedlovszky*,‡,|,⊥ UniVersity Chemical Laboratory, UniVersity of Cambridge, Lensfield Road, Cambridge CB2 1EW, England, U.K., HAS Research Group of Technical Analytical Chemistry, Szt. Gelle´rt te´r 4, H-1111 Budapest, Hungary, Department of Inorganic and Analytical Chemistry, Budapest UniVersity of Technology and Economics, Szt. Gelle´rt te´r 4, H-1111 Budapest, Hungary, EKF Department of Chemistry, Leańyka utca 6, H-3300 Eger, Hungary, and Laboratory of Interfaces and Nanosize Systems, Institute of Chemistry, Eo¨tVo¨s Lorańd UniVersity, Pa´zmańy P. Stny 1/A, H-1117 Budapest, Hungary ReceiVed: September 27, 2010; ReVised Manuscript ReceiVed: October 29, 2010

The properties of the interface between water and benzene are investigated in detail on the basis of 23 Monte Carlo computer simulations performed at various temperatures and pressures. The interfacial properties are analyzed in terms of the novel identification of the truly interfacial molecules (ITIM) method, by which the intrinsic (i.e., capillary wave corrugated) surface of the two phases can be detected. The obtained results show that the use of a simple, nonintrinsic definition of the interface (made on the basis of the average density profiles of the components) not only leads to a systematic error in determining the list of the truly interfacial molecules, but this error is also reflected in the erroneous calculation of the thermodynamic properties of the system. The obtained results show that the reciprocal width and reciprocal amplitude of both surface layers decrease linearly with the temperature and reach the value of zero (i.e., the corresponding parameters become infinite) at the point of mixing of the two phases. Similar linear relation is observed between these reciprocal quantities and the logarithm of the pressure, but only above a certain temperature. This temperature is thought to be the upper end of the lower critical line of the phase diagram of the system; however, any reliable support of this conjecture would require a considerably larger number of simulations in the temperature range close to this line. The orientational preferences of the surface water molecules, governed by the principle of maximizing the number of their hydrogen-bonded neighbors, are found to be insensitive to the thermodynamic conditions but become weaker with increasing temperature and decreasing pressure. The lateral hydrogen-bonding network of the surface water molecules, spanning the entire water surface at ambient conditions, is found to undergo a percolation transition well, i.e., 200-400 K below the mixing of the two phases, indicating that the existence of such a percolating lateral network is not a universal feature of the water surface but depends also on the thermodynamic conditions. 1. Introduction Intensive experimental investigation of the molecular level properties of fluid (i.e., liquid-liquid and liquid-vapor) interfaces started in the past 2 decades, as a consequence of the appearance and spreading of various methods that are able to selectively probe molecules located at the boundary of two phases, such as vibrational sum frequency generation (SFG) and second harmonic generation (SHG) spectroscopy or X-ray and neutron reflexivity.1-19 Further, simultaneously with this rapid development of suitable experimental techniques more and more powerful computers became widely used, giving rise, among others, to computer simulation investigations of interfacial systems. Computer simulation studies can very well complement experimental investigations, since in simulations full, threedimensional insight into the system studied can be obtained at * To whom correspondence should be addressed. E-mail: pali@ chem.elte.hu. † University of Cambridge. ‡ HAS Research Group of Technical Analytical Chemistry. § Budapest University of Technology and Economics. | EKF Department of Chemistry. ⊥ Eo¨tvo¨s Lorańd University.

the molecular level, whereas the models used in the simulations have to be validated against experimental data. Not surprisingly, the wealth of experimental investigations of fluid interfaces is also accompanied by a rapidly growing number of computer simulation studies of such systems.20-56 In simulating fluid interfaces at atomistic resolution, however, one has to face the problem that finding the exact location of the interface is far from being a trivial task. This problem is related to the fact that on the atomistic length scales the liquid surface is corrugated by capillary waves. In the majority of the interfacial simulations, this problem is simply neglected, and the region of the interface is defined as a layer between two planes parallel with the macroscopic interfacial plane, between which the densities of the components are between the values characteristic to the two bulk phases. However, the presence of such a layer of intermediate densities is the consequence of the fact that in this region both phases are present due to the rugged nature of the interface separating them, and hence, the calculation of any physical property in this slab inevitably involves averaging over both phases. In other words, the neglect of the capillary waves in locating the interface leads to the misidentification of a large (and unknown) number of particles, both

10.1021/jp109227j  2010 American Chemical Society Published on Web 11/17/2010

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as interfacial and as noninterfacial ones. This misidentification results in a systematic error of unknown magnitude in the calculation of any interfacial property. Unfortunately, in spite of this source of systematic error, this treatment of the interface in fluid systems became now rather widespread in computer simulation analyses. In spite of its present ignorance, the problem of the capillary waves was already noticed in the very first simulation studies of fluid interfaces. To circumvent the problem, Linse20 and Benjamin21 divided the system into several slabs using planes parallel to the interface normal and determined the position of the interface in each slab separately. This method, applied later to a number of systems,24,27-29,31 suffers, however, from the problem that the resulting intrinsic interface strongly depends on the number of slabs used. One possible way of circumventing this problem is to determine the two-dimensional Voronoi tessellation of the molecules projected to the macroscopic plane of the interface, as suggested by Berkowitz et al.57 Recently the original method was further elaborated by Jorge and Cordeiro,45 who determined the number of slabs required for convergence to the intrinsic interface. This method of grid-based intrinsic profiles (GIP) has been applied for various interfacial systems.45,46 Chowdhary and Ladanyi used a criterion based on the vicinity of the molecules of the other phase to identify molecules located right at the surface of their own phase.41 Although the use of this method, called surface layer identification (SLI), is naturally limited to liquid-liquid interfaces, its extended version (SLIx) is shown to be practically identical with other intrinsic methods.58 In their intrinsic sampling method (ISM) Chacoń and Tarazona proposed a different approach, namely to determine the surface of minimum area going through a set of pivot atoms.59 The ISM method was later further refined60 and applied to a number of different systems.38,43,48 Since the ISM method is based on a self-consistent algorithm,59 it can naturally find the molecules located right at the surface of their phase; however, the self-consistency of the algorithm makes it computationally rather demanding.58 Wilard and Chandler proposed recently an intrinsic method which, unlike the other similar methods, does not rely on the assumption that the interface is macroscopically flat, and hence, it can be used for interfaces of various (even irregular) shapes.61 Recently we proposed a new intrinsic method for the identification of the truly interfacial molecules (ITIM).47 The ITIM method finds the molecules located at the surface of their phase by moving a probe sphere of a given radius along a number of test lines perpendicular to the macroscopic plane of the interface, starting from the bulk of the opposite phase. Once the probe sphere touches the first molecule of the phase to be studied, it is stopped, and the molecule touched is identified as being interfacial. When the probe sphere is moved along all the test lines, the full set of the truly interfacial molecules is identified.47 Further, by disregarding the set of molecules identified as interfacial and repeating the entire procedure, the list of molecules forming the subsequent molecular layers beneath the surface of their phase can also be identified. The ITIM method was shown to favorably compare with other intrinsic methods, both in terms of accuracy and computational efficiency,58 and was successfully applied to a number of liquid-vapor47,50,51,53-55 and liquid-liquid interfaces.49,52,56 Although the majority of the intrinsic methods are based on rather simple algorithms, they always assume that for each molecule it is a priori known to which phase it belongs. In the majority of the interfacial simulation studies, such an assumption is justified by the fact that, due to the limited size of the system,

Pa´rtay et al. practically no molecule of one phase can penetrate into the other phase. However, in studying interfaces between two phases, both of which contain also molecules of the other phase as a minor component, i.e., when the mutual solubilities of the two components are not negligible even in systems of size typical in computer simulations, the determination of the phase to which a particular molecule belongs is not always an obvious task. In particular, it has to be decided for a number of molecules whether they are located at the surface of their own phase, forming ripples of that, or they are already dissolved in the other phase, although being close to the interface. This problem prevented researchers so far from studying the interface of two mixed liquid phases by intrinsic surface analysis methods. In this paper, we propose a method to circumvent this problem and apply it to the analysis of the water-benzene interface in a broad range of thermodynamic states ranging from ambient conditions (where the molecules of the two phases practically do not mix with each other, at least in small systems used in computer simulations) up to the vicinity of the upper critical line of the phase diagram (where differences between the two phases vanish and the two liquid phases disappear as their components fully mix with each other). The proposed algorithm is based on finding physically relevant connectivity criteria of the like molecules, by which it can be decided whether two molecules are mutual neighbors of each other or not. Then the largest cluster consisting of like molecules linked together by chains of mutual neighbors is determined, and this cluster is regarded as the bulk phase of the given component, whereas the small oligomers not belonging to this largest cluster are regarded as being dissolved in the other phase. The water-benzene system has been widely studied both by experimental62-71 and computer simulation methods.20,39,40,72-76 Mutual solubilities of the two components in each other have been measured at various high temperature and high pressure states a number of times in the past 50 years.62,63,65-67,69,70 The phase diagram of this system was also determined already in the 1960s.64 Similarly to other two-component systems, this phase diagram also contains two critical lines. The lower critical line, terminating in the critical point of neat benzene at 562.6 K and 49.2 bar,77 separates the two-phase liquid and the three-phase (i.e., two liquids and a vapor) systems. The upper critical line, starting from the critical point of water at 647.1 K and 220.55 bar,78 separates two-phase liquid and one-phase systems. This upper critical curve first goes to lower pressures and temperatures, but at the critical mixing temperature, estimated to be 570-57563 and 567 K,64 it changes slope and turns back to higher pressures and temperatures.64 The experimental critical curves of the water-benzene phase diagram are shown in Figure 1. Water-benzene systems have also been studied by computer simulation methods. The very first simulation of the liquid-liquid interface, done by Linse more than 20 years ago, already targeted the interface between water and benzene.20 Hydrophobic hydration of a benzene molecule was simulated by Ravishanker et al.72 and by Linse.73 Dang and Feller calculated the solvation free energy profile of a single benzene molecule across the liquid-vapor interface of water.74 Nieto-Draghi et al.75 and Ikawa76 studied one-phase water-benzene mixtures at several thermodynamic state points above the upper critical curve. In an early study, we analyzed the orientation of the water molecules near the water-benzene interface at several thermodynamic state points.39 Later, we presented a detailed investigation of this system in a broad range of thermodynamic states from ambient to supercritical conditions.40 In this study, we

ITIM Analysis of the Water-Benzene Interface

Figure 1. Phase diagram of the water-benzene system, showing the thermodynamic state points 1-23 at which the simulations were performed. The experimental lower and upper critical curves are shown by dashed lines.64 These curves terminate at the experimental critical point of neat benzene77 (square) and water78 (triangle). The simulated upper critical curve obtained in a previous, nonintrinsic analysis40 is shown by a solid line. The gray shaded area indicates the correct location of the simulated upper critical curve, as resulted from the present, intrinsic analysis. The striped region shows the location of the percolation threshold of the lateral hydrogen-bonding network of the surface water molecules.

performed Monte Carlo simulations in 23 thermodynamic state points, and besides analyzing various properties of the interface, we also estimated the location of the upper critical line of the phase diagram.40 However, in this study, the interface was still analyzed in the conventional, nonintrinsic way, and hence, the results were affected by the aforementioned systematic error of unknown magnitude introduced by the nonintrinsic treatment of the interface. In this paper we reanalyze the results of our previous simulations40 by means of the ITIM method, defining the bulk phase of the components as their largest cluster in the basic simulation box and hence distinguishing between molecules located in their own phase and those being dissolved in the other phase. In this way, ITIM analysis always targets the surface of the largest continuous cluster of the molecules, while isolated patches of molecules are disregarded. We analyze the width and roughness of the surface layers as well as the average distance of the surface layers of the two liquid phases. Particular attention is paid to the dependence of these properties on the thermodynamic conditions and to their relation with the critical line of mixing (upper critical line). Further, the temperatureinduced breakdown of the lateral percolating hydrogen-bonding network of the surface water molecules is also thoroughly investigated. The study is completed by the analysis of the surface orientation of the truly interfacial water molecules. 2. Computational Details 2.1. Monte Carlo Simulation. Monte Carlo simulations of the water-benzene system have been performed on the isothermal-isobaric (N, p, T) ensemble at 23 different thermodynamic state points. The temperature and pressure corresponding to the different simulations are summarized in Table 1. The simulations have already been described in detail in our previous publication;40 thus, their details are only briefly reminded here. Each system has consisted of 536 water and 108 benzene molecules, described by the SPC/E potential79 and a six-site Lennard-Jones potential model,80 respectively. According to these models, both water and benzene molecules have been kept rigid in the simulations.

J. Phys. Chem. C, Vol. 114, No. 49, 2010 21683 The simulations have been done using the program MMC.81 Every 500 particle displacement (i.e., translation and rotation) moves have been followed by a volume change attempt. The maximum translation and rotation of a displaced water molecule have been 0.25 Å and 15°, respectively, whereas for benzene these values have been 0.4 Å and 25°, respectively. The maximum change of the system volume in one step has been set to 400 Å3. In the volume change steps only the length of the box edge perpendicular to the macroscopic plane of the interface, X, have been altered, while the Y and Z edge lengths have been kept unchanged at their initial value of 25.216 Å. In this way, at least 20% of the attempted moves have been successful in each system for each type of Monte Carlo steps. The systems have been equilibrated by performing 5 × 108 Monte Carlo steps. Then, in the production stage of the simulations, 1000 equilibrated sample configurations per system separated by 106 Monte Carlo steps each have been saved for further evaluation. The sample configurations have been translated along the interface normal axis X in such a way that the center of mass of the 536 water molecules has been placed to the middle of the basic box, i.e., at X ) 0 Å. The two interfaces present in the basic box have been treated separately, and all results have been averaged not only over the 1000 sample configurations but also over these two interfaces per configuration. 2.2. ITIM Analysis. In ITIM analysis the set of the truly interfacial molecules of the phase of interest is searched for by moving a probe sphere of given radius along a set of test lines from the bulk of the opposite phase. Once the probe sphere touches the first atom of the phase studied, it is stopped, and the molecule to which the touched atom belongs is marked as interfacial. Once the probe sphere has been moved along all the test lines, the full list of the truly interfacial molecules is determined. Further, the set of intersection points of the probe sphere surface with the test line at the position where the sphere is stopped provides an estimate of the covering surface (i.e., intrinsic surface) of the phase of interest. (Obviously, among the two intersection points of the probe sphere surface and the test line always the one closer to the analyzed phase should be considered.) The list of the truly interfacial molecules found obviously depends on the radius of the probe sphere used. However, the use of such a free parameter seems to be inevitable in any kind of intrinsic surface analysis.58 Further, in analyzing the interface between two liquid phases it seems to be advantageous to use the same probe sphere for scanning both of the liquid surfaces. We have previously shown that at the water surface the list of the truly interfacial molecules found and the properties of the intrinsic surface detected depend only rather weakly on the probe sphere radius around its value of 2 Å.47 Further, it has recently been demonstrated by Jorge et al. that, in general, the optimal probe sphere radius to be used should be close to half the position of the first peak of the pair correlation function characteristic of the phase studied.58 Since the first peak of the pair correlation function of the benzene CH groups is located at 3.9 Å, we have used the probe sphere radius value of 2.0 Å for both phases in the present study. In detecting the point where the probe sphere is stopped, the diameter of the different atoms have been approximated by their respective Lennard-Jones distance parameters σ (i.e., 3.166 Å for the water O atom79 and 3.37 Å for the benzene CH groups80). It is also clear that the detection of all truly interfacial molecules requires the use of sufficiently large number of test lines. It has been demonstrated by Jorge et al.58 for water that a grid of test lines with a grid spacing of 0.5 Å is sufficient for

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TABLE 1: Properties of the Different Systems Simulated water surface

benzene surface

state point

T/K

p/bar

ξ

a/Å

δ/Å

ξ

a/Å

δ/Å

D/Å

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

300 375 450 300 375 450 300 375 450 575 650 800 1200 450 575 650 800 1200 450 575 650 800 1200

1 2 10 100 100 100 1000 1000 1000 1000 1000 1000 1000 2500 2500 2500 2500 2500 5000 5000 5000 5000 5000

0.721 0.788 0.908 0.670 0.777 0.879 0.704 0.748 0.816 1.009

1.972 2.416 3.195 1.992 2.469 3.102 1.918 2.242 2.714 3.773

2.82 3.34 4.21 2.82 3.39 4.09 2.79 3.14 3.70 5.14

0.766 0.898

2.149 2.803

4.15 5.13

4.77 5.51

0.765 0.856 1.035 0.708 0.768 0.841 1.023 1.241

2.096 2.764 3.742 1.999 2.326 2.842 4.136 6.048

4.08 4.92 6.41 3.92 4.28 4.88 6.44 8.85

4.69 5.31 6.45 4.39 4.66 4.99 5.78

0.782 0.932 1.041

2.511 3.352 4.185

3.47 4.55 5.57

0.773 0.901 1.019

2.596 3.457 4.485

4.46 5.54 6.58

4.49 4.80 5.02

0.810 0.865 0.964

2.280 3.028 3.584

3.34 4.21 4.75

0.723 0.821 0.908 1.190

2.371 3.139 3.657 6.165

4.29 5.03 5.59 9.00

4.15 4.31 4.37

this purpose. Such a grid spacing seems to be sufficient also for the benzene phase, as the CH group of the benzene molecule is somewhat larger than the water O atom. Therefore, we have used a grid of 50 × 50 test lines for the analysis of both liquid surfaces here. An important problem of the ITIM analysis performed here concerns the nonzero mutual solubilities of the two components in each other. Thus, at the high temperature state points the number of water and benzene molecules penetrating into the opposite phase is not zero even in the small sample contained by the basic simulation box.40 The molecules penetrating to the bulk of the opposite phase can then stop the probe sphere before it reaches the region of the interface, leading to their misidentification as interfacial molecules. Further, these molecules also make the parts of the surface of their own phase located behind them unreachable for the probe sphere, and hence, the truly interfacial molecules located at these “shaded” parts of the surface cannot be seen by the ITIM analysis. It should be noted that no such problem occurred in the previous ITIM analyses reported, since in these former studies the solubility of the components in the opposite phase was always low enough that the number of molecules penetrating into the opposite phase in the small sample of the simulation was practically zero.47,49-56 In order to overcome this problem, here we have adopted the following procedure. We have detected the largest continuous cluster of both types of molecules and regarded these largest clusters as the two bulk liquid phases. Clusters have been defined as the full set of molecules among which any two are connected to each other via a chain of contact pairs. Two benzene molecules have been regarded as being in contact with each other if the distance of their nearest CH groups was less than 4.25 Å, whereas two water molecules have been regarded as a contact pair if their O atoms were closer than 3.35 Å. These limiting distances correspond to the first minimum positions of the corresponding pair correlation functions. In this way, molecules or even small clusters that are disconnected from (i.e., lost contact with) their bulk liquid phase, and hence dissolved in the opposite phase, have been identified and disregarded in the ITIM procedure. The molecules forming the surface layers

of both phases and the second molecular layer beneath the surface of the water phase identified this way in instantaneous configurations at two different thermodynamic state points are illustrated in Figure 2. 3. Results and Discussion 3.1. Density Profiles. The density profiles of the water and benzene molecules along the macroscopic normal axis X of the

Figure 2. Instantaneous snapshot of the surface layer of the benzene molecules (gray) as well as the surface layer (red) and second layer beneath the surface (green) of the water molecules as taken out from the simulations performed (a) at state point 7, i.e., at 300 K and 1000 bar, and (b) at state point 12, i.e., at 800 K and 1000 bar.


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Figure 3. Density profile of water (open circles) and benzene (filled circles) along the macroscopic interface normal axis X as obtained at three different thermodynamic state points. The density profiles of the water and benzene molecules constituting the surface layer of their phase as well as of water molecules belonging to the second molecular layer beneath the surface are also indicated (solid, dashed, and dash-dotted lines, respectively).

interface are shown in Figure 3 at three different thermodynamic state points. As is seen and has been discussed in detail in our previous publication, the profiles increasingly overlap with increasing temperature. The increasing extent of the region of intermediate compositions along the interface normal axis X suggests that the interface becomes more rugged with increasing temperature. The problem of the pressure and temperature dependence of the interfacial roughness is addressed later in this paper. Figure 3 also shows the density profiles of the water and benzene molecules constituting the first molecular layer of the respective phases (as obtained from ITIM analysis) and also that of the water molecules belonging to the second molecular layer beneath the interface (as obtained from an ITIM analysis performed after removal of the first layer water molecules). As is seen, the density peaks of the interfacial (first layer) molecules extend well beyond the X values at which the overall density profile of the same component reaches the bulk phase value. Further, the water molecules constituting the second subsurface layer give a considerable contribution to the water density profile even in the X range of intermediate water densities. These findings clearly stress the importance of using an intrinsic analysis of the interfacial molecules, since a rather large number of molecules would be misidentified as interfacial or as noninterfacial ones on the basis of the conventional, nonintrinsic definition of the interface based simply on the average density profiles of the molecules in every case. The temperature and pressure dependence of the distribution of the truly interfacial molecules along the macroscopic interface normal axis X is presented in Figure 4, showing the density profiles of both surface layers at different temperatures at the pressure of 1000 bar and also at different pressures at the temperature of 450 K. As is seen, these distributions become progressively broader with increasing temperature and decreas-

Figure 4. Density profile of the water (top panels) and benzene (bottom panels) molecules constituting the surface layer of their phase as obtained (a) from the simulations performed at different temperatures at the pressure of 1000 bar and (b) from those at different pressures at the temperature of 450 K.

ing pressure. Further, the overlap of the two density peaks is, in general, larger at higher temperatures. In order to analyze the dependence of the surface layer properties on the thermodynamic conditions in a more quantitative way, we have fitted a Gaussian function to the obtained density profiles at every thermodynamic state considered. As is seen from Figure 4, the density data can be well-fitted by a Gaussian function in most of the cases; however, in some cases (e.g., at T ) 650 K, p ) 1000 bar in the case of the water and at T ) 450 K, p ) 10 bar in the case of the benzene surface layer) the density profiles are seen to deviate considerably from the Gaussian shape. To avoid fitting a Gaussian function to data that are not following a Gaussian distribution, we plotted in Figure 5 the correlation coefficient R of the density data and their best fitting Gaussian function for both water and benzene in all thermodynamic state points where a two-phase system was obtained. As is seen, the correlation coefficients agree quite well with each other in the majority of the state points considered, being generally somewhat larger for the water than for the benzene surface. This better agreement of the water than benzene density peaks with the Gaussian shape is related to the smaller size of the water molecule and indicates that probably a slightly more accurate mapping of the benzene surface would have been achieved by optimizing the ITIM

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Figure 5. Correlation coefficient of the Gaussian fit of the density profiles of the water (open circles) and benzene (filled circles) molecules constituting the surface layer of their phase, as obtained from the simulations at all the thermodynamic states where a two-phase system was obtained. Encircled are the state points corresponding to poor quality fits; the properties of the corresponding surface layers are omitted from the following analyses.

parameters (e.g., grid spacing, probe sphere size) for this surface. It is also seen, however, that in a few thermodynamic states, i.e., at state points 11 (T ) 650 K, p ) 1000 bar), 17 (T ) 800 K, p ) 2500 bar), and 22 (T ) 800 K, p ) 5000 bar) for the water and in state points 3 (T ) 450 K, p ) 10 bar) and 17 (T ) 800 K, p ) 2500 bar) for the benzene surface, significantly worse fits were obtained. The reason for this deviation from the Gaussian shape at certain, high-temperature or low-pressure thermodynamic states is clearly nonphysical. In some cases it can be explained by the finite size effect, namely, that the distributions of the two surface layers of a given phase that are present in the finite basic simulation box are broad enough to overlap with each other. Another reason for this non-Gaussian behavior can originate from the nonzero mutual solubilities of the unlike molecules in each other. Thus, if a relatively large water cluster, being isolated from the aqueous phase, is located beyond the benzene surface, then it may result in a long, slowly decaying tail of the interfacial benzene density peak extending toward the water phase. Although the mutual solubility of the molecules is a real physical effect, the proper sampling of the systems in such cases is clearly questionable, again because of the limited system size. Considering also the results of a recent theoretical study that supported the idea that the density distribution of the surface molecules obeys Gaussian statistics,82 we have omitted the surfaces deviating strongly from the Gaussian density distribution (encircled in Figure 5) from the following analyses. Having a Gaussian fitted to the surface density distributions, the width of the surface layer of either water or benzene can be characterized by the width of the fitting Gaussian function at half-maximum, δ. The value of δ may increase if the roughness of the corresponding surface layer increases; however, it can also be due to the increasing penetration of the surface layer into the bulk region of the system, i.e., to the decreasing order of the layering structure of the corresponding phase beneath its surface. On the other hand, the width of the interfacial region can be characterized by the distance D between the peak positions of the density distributions of the water and benzene surface layers.52,56 Besides the increasing roughness of the two

Pa´rtay et al. surfaces, this quantity may also increase due to the increasing separation of the two phases, i.e., the broadening of the void space between them. The δ values characterizing the width of the two surface layers as well as the D values characterizing the interfacial width are collected in Table 1 as obtained in all cases when the corresponding surface densities are sufficiently close to the Gaussian shape (see Figure 5). As a general conclusion, we can say that both the two δ values and D become larger with increasing temperature (i.e., approaching the upper critical line of the system) and decreasing pressure. A more detailed analysis of the temperature and pressure dependence of these properties is given in a following subsection. 3.2. Surface Roughness. Since, besides the full list of the truly interfacial molecules, the ITIM procedure also results in a large number of points that approximate the covering surface of the corresponding phase (see Figure 1 of ref 47), it provides a unique opportunity also to quantify the molecular scale roughness of this surface.47,53 However, even if the covering surface of a phase is known, an unambiguous characterization of its roughness is a nontrivial task. Clearly, such a characterization requires the use of at least two independent parameters, i.e., a frequency-like and an amplitude-like one. For determining such a parameter pair, recently we proposed the following method.53 The average normal distance of two surface points, dj (i.e., their distance along the macroscopic surface normal axis X), exhibits a saturation curve as a function of their lateral distance, l (i.e., their distance within their projections to the macroscopic plane of the surface, YZ). At small enough l values, the value of dj rises linearly with l, the slope of this linear rise being ξ, whereas at large enough lateral distances the dj(l) function converges to a saturation value a. The frequency- and amplitude-like roughness parameters, ξ and a, respectively, can simply be determined by fitting the function

dj )

aξl a + ξl

(1)

to the obtained dj(l) data. The ξ and a roughness parameters are collected in Table 1 for all surfaces for which the density distribution is sufficiently close to the Gaussian shape (see Figure 5), whereas the temperature and pressure dependence of the dj(l) roughness curve is illustrated in Figure 6 for both surfaces. As is seen, the frequency-like parameter ξ depends only rather weakly on the thermodynamic conditions, in particular, on the pressure (although the increase of the temperature or decrease of the pressure leads to a slight increase of it). On the other hand, the temperature and pressure dependence of the amplitude parameter a is quite clear. Thus, similarly to δ and D, this parameter also increases with increasing temperature and decreasing pressure. A detailed analysis of the temperature and pressure dependence of a along with that of the parameters δ and D is presented in the following subsection. 3.3. Temperature and Pressure Dependence of the Surface Properties. In order to study the temperature and pressure dependence of various surface properties, we have plotted the variation of the roughness parameter a of the benzene and water surfaces, the average width of the two surface layers, δ, and the average interfacial width, D, with the temperature along the 100, 1000, 2500, and 5000 bar isobars and with the pressure along the 300, 375, 450, and 575 K isotherms (and, for the properties related to the surface of the benzene phase, also at 650 K) in Figure 7. As is seen, all these parameters increase progressively with increasing temperature along each isobar,



Figure 6. Average normal distance of two points of (a) the water and (b) the benzene surface (i.e., their distance along the macroscopic interface normal axis X) as a function of their lateral distance (i.e., their distance in the YZ plane), as obtained from the simulations performed at different pressures at the temperature of 450 K (top panels) and from those at different temperatures at the pressure of 1000 bar (bottom panels).

Figure 7. (a) Temperature and (b) pressure dependence of the amplitude parameter of the surface roughness, a (top panels); the width of the surface layers, δ (middle panels); and the interfacial width, D (bottom panels). For the definition of the parameters a, δ, and D, see the text. The quantities corresponding to the water and benzene surfaces and to the interface between the two phases are shown by open, filled, and gray symbols, respectively.

suggesting that they probably diverge at the corresponding mixing temperature. Also, all of these properties are seen to decrease with increasing pressure, although this decrease is usually rather weak, in particular, at lower temperatures. However, since the system cannot cross the upper critical line along isotherms that are below the critical mixing temperature, no divergence of these curves is expected, at least at lower temperatures. To examine the possibility of divergence of the a, δ, and D values with increasing temperature along the four isobars considered, we also plotted their reciprocal values against the temperature along each of these isobars (see Figure 8). As is

seen, the temperature dependence of a-1, δ-1, and D-1 can always be very well fitted by a straight line. The extrapolation of these fitted lines to zero value provides an estimate for the temperature of divergence of these values corresponding to the given isobar. The estimated temperatures of divergence of all these parameters along all the four isobars are collected in Table 2. As is seen, the temperatures obtained considering the two a and the two δ values agree reasonably well, i.e., within an error bar never larger than (75 K with each other, whereas the value obtained from the extrapolation of the D-1(T) data strongly deviates from them in every case. This is understandable considering the physical meaning of the a, δ, and D parameters.

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Figure 8. Temperature dependence of the reciprocal amplitude parameter of the surface roughness, a-1 (top panel); reciprocal width of the surface layers, δ-1 (middle panel); and reciprocal interfacial width, D-1 (bottom panel), along the 5000 bar (squares), 2500 bar (circles), 1000 bar (up triangles), and 100 bar (down triangles) isobars. The straight lines fitted to the data along each isobar are shown by dashed lines. For the definition of the parameters a, δ, and D, see the text. The quantities corresponding to the water and benzene surfaces and to the interface between the two phases are shown by open, filled, and gray symbols, respectively.

TABLE 2: Critical Mixing Temperatures of the Simulated Water-Benzene System (in K), As Obtained from the Temperature Dependence of Various Properties at Different Pressures p/bar

awat

aben

δwat

δben

D

D10-90 a

5000 2500 1000 100

992 950 854 716

1019 929 825 638

1113 980 900 781

1138 1073 946 714

4346 2349 1445 853

984 900 716 607

a

10%-90% width of the nonintrinsic interface; see ref 40.

Thus, divergence of a and δ indicates that the corresponding surface becomes infinitely rough and broad, respectively, which means that the mixing of the two phases occurs. Therefore, the divergence temperatures of these quantities can be regarded as estimates for the mixing temperature corresponding to the given isobar. The estimated temperature range within which the upper critical line of the system is located is also indicated in Figure 1 as obtained from this analysis. On the contrary, D measures the average separation of the two surfaces, and this quantity is not expected to diverge at the point where the two phases mix with each other. In other words, while the temperature-induced increase of the surface roughness parameter a and of the surface layer width δ reflects the increasing penetration of the two phases into each other, the increase of D with the temperature can simply be related to the decreasing density of the system due to the increasing thermal motion of the molecules. Therefore, no particular physical meaning is attributed to the estimated divergence temperature of D. It is also evident from Figure 1 and Table 2 that the critical mixing temperatures estimated from the temperature dependence

Figure 9. Dependence of the reciprocal amplitude parameter of the surface roughness, a-1 (top panel); reciprocal width of the surface layers, δ-1 (middle panel); and reciprocal interfacial width, D-1 (bottom panel), on the logarithm of the pressure along the 300 K (squares), 375 K (circles), 450 K (up triangles), 575 K (down triangles), and 650 K (diamonds) isotherms. The straight lines fitted to the data along the high temperature isotherms are shown by dashed lines. For the definition of the parameters a, δ, and D, see the text. The quantities corresponding to the water and benzene surfaces and to the interface between the two phases are shown by open, filled, and gray symbols, respectively.

of either a or δ of any of the two surfaces are consistently larger than what was previously estimated in a nonintrinsic analysis from the temperature dependence of the 10%-90% width of the average water density profile.40 This finding again stresses the importance of analyzing the intrinsic surface of fluid phases instead of simply defining their surface through the variation of the average density of its components, as it clearly demonstrates how the systematic error originating from the incorrect identification of the interfacial and noninterfacial molecules leads to a systematic error of the calculated thermodynamic properties of the system. Finally, it should be noted that the calculated upper critical line of the system goes at considerably higher temperatures than the experimental line64 (see Figure 1). The reason of this deviation is, as it has already been pointed out earlier,40 that the used simple model of benzene is not able to describe the hydrogen-bonding-like attractive interaction occurring between the delocalized π electron system of the benzene ring and the H atom bearing fractional positive charge of a nearby water molecule.70 The lack of this cohesive force in the model leads to the observed shift of the mixing of the two phases to higher temperatures. To find a simple functional relation between the surface parameters a, δ, and D and the pressure of the system, we plotted the reciprocal values of these parameters against the logarithm of the pressure in Figure 9. As is seen, the a-1(ln p), δ-1(ln p), and D-1(ln p) data can be well-fitted by straight lines at sufficiently high temperatures (i.e., from 450 K in the case of the benzene surface related properties and D and at 575 K in the case of the water surface related properties), whereas at lower



Figure 10. (a) Definition of the local Cartesian frame fixed to the individual water molecules and that of the polar angles ϑ and φ used in the analysis of surface orientation. (b) Orientational maps of the water molecules constituting the entire surface layer (first column) as well as its regions C-A (second-fourth columns, respectively) at five different pressures at T ) 450 K and (c) at four different temperatures at p ) 1000 bar. For the definitions of the regions A-C, see the text. Lighter colors indicate higher probabilities. (d) Illustration of the found preferred orientations of the surface water molecules. X is the macroscopic surface normal vector pointing, by convention, from the water to the benzene phase.

temperatures no such linear dependence is seen, and the dependence of a-1, δ-1, and D-1 on ln p is much weaker than along the higher temperature isotherms. The observed behavior suggests that the linear dependence of the reciprocal surface parameters considered on ln p exists probably in the temperature range above the point where the lower critical line of the system terminates. However, this conjecture cannot be confirmed on the basis of the present set of simulations. 3.4. Orientation of the Surface Waters. The accurate identification of the molecules that are located right at the surface of their phase, achievable by the use of the ITIM method, also enables us to investigate the orientational preferences of the surface molecules relative to the macroscopic plane of the surface. However, since our results revealed that the orientation of the benzene molecules forming the surface layer of their phase is almost completely random in every thermodynamic state considered (data not shown), here we limit the discussion of the surface orientation of the molecules to the aqueous phase of the system.

The orientation of a rigid body (e.g., an SPC/E water molecule) relative to an external direction (e.g., the macroscopic surface normal vector, X, pointing, by our convention, from the aqueous to the benzene phase) can be fully described by two independent orientational variables, such as the polar angles ϑ and φ of the vector X in a local Cartesian frame fixed to the water molecule. Thus, the complete description of the orientational statistics of the surface water molecules relative to the surface normal vector X requires the calculation of the bivariate distribution of these orientational variables.33,36 Here we define the local Cartesian frame in the following way (see Figure 10a). Its axis x is perpendicular to the molecular plane; axis z coincides with the main symmetry axis of the molecule, and it is directed from the O atom toward the H atoms, whereas axis y is perpendicular to the above two. Due to the C2V symmetry of the water molecule this frame can always be chosen in such a way that the polar angle φ does not exceed 90°. Further, since ϑ is an angle between two general spatial vectors, but φ is an angle formed by two vectors restricted to lay in a given plane

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(i.e., the xy plane of the local frame), uncorrelated orientation of the water molecules with the surface results in uniform bivariate distribution only if cos ϑ and φ are chosen to be the two orientational variables.33,36 The P(cos ϑ,φ) bivariate distributions of the surface water molecules are shown in parts b and c in Figure 10 as obtained at various pressures along the 450 K isotherm and at various temperatures along the 1000 bar isobar, respectively. In order to investigate also the influence of the local curvature of the water surface on the orientational preferences of the surface water molecules, we have defined three regions, denoted by A, B, and C, respectively, within the surface layer of water and calculated the P(cos ϑ,φ) bivariate orientational distributions also in these regions separately. The resulting orientational maps are also shown in Figure 10. Regions A and B extend from the X values at which the density of the surface molecules drops to 10% and 50% of its maximum value, respectively, toward the benzene phase, whereas region C, located at the other, aqueous side of the surface water density peak, extends from the point where the density of the surface waters is 50% of the maximum value toward the aqueous phase. Thus, regions C and B cover the troughs and crests, respectively, whereas region A covers only the tips of the crests of the molecularly wavy water surface. In other words, typically in region C the surface is locally of concave curvature, whereas in regions A and B it is locally of convex curvature. As is seen from Figure 10, the dominant orientation of the surface water molecules corresponds to the cos ϑ and φ values of 0 and 0°, respectively, at every state point considered. In this orientation, denoted by I, the water molecule lies parallel with the macroscopic plane of the surface YZ (see Figure 10d). Apart from the state point at 575 K, this orientation is found to be preferred also in regions B and C of the surface layer in every case. However, it is also seen that in region B the corresponding peak of the P(cos ϑ,φ) orientational map is shifted to slightly negative values, whereas in region C it is shifted to slightly positive cos ϑ values. This small shift indicates a slight tilt of the plane of the water molecule in its preferred alignment relative to the macroscopic surface plane YZ. This tilt is such that in region B the two H atoms (and also one of the two lone pair directions) of the water molecule point slightly to the aqueous and the other lone pair straight to the opposite phase, whereas in region C the two H atoms and one of the lone pairs point flatly to the benzene, while the other lone pair points straight to the water phase. It is also seen that besides orientation I another orientation, characterized by the cos ϑ ≈ 0.5 and φ ) 90° values, is preferred in region B, and a third orientation, corresponding to the {cos ϑ ≈ -0.5; φ ) 90°} point of the orientational map is preferred in region C. These orientations, marked by II and III, respectively, are also illustrated in Figure 10d. In both of these orientations the plane of the water molecule is perpendicular to the macroscopic surface plane YZ. One of the O-H bonds points almost straight to the benzene phase in orientation II and to the aqueous phase in orientation III, while the other O-H bond, as well as the two lone pairs, points flatly to the water phase in orientation II and to the benzene phase in orientation III. Further, in region A clearly orientation II is the dominant one, and at high temperatures or low pressures the preference of the molecules for orientation I is already lost here. The observed orientational preferences are rather similar to what has been previously observed at various water-apolar interfaces under ambient thermodynamic conditions,56 and can be explained in the following way. At positions of locally

Pa´rtay et al. convex surface curvature (i.e., in regions A and B) the water molecules prefer orientations in which they sacrifice one of their four potential hydrogen-bonding directions (i.e., one of the two lone pair directions in the tilted orientation I and one of the O-H bond directions in orientation II), and on this expense, they can maintain three strong hydrogen bonds along the other three possible directions with the neighboring waters. However, at positions of locally concave surface curvature (i.e., in region C), the water molecules can maintain all their four possible hydrogen bonds with their neighbors if they straddle three of the four possible hydrogen bond directions (i.e., two O-H and one lone pair directions in the tilted orientation I and two lone pair and one O-H directions in orientation III) by the locally concave surface. It should be noted that similar principles were found to govern the orientational preferences of the water molecules hydrating apolar spherical solutes of varying size.83 In investigating the temperature and pressure dependence of the observed orientational preferences, it is seen that the preferred orientations themselves are rather insensitive to the thermodynamic conditions, as the position of the peaks of the P(cos ϑ,φ) orientational maps remains unchanged with varying pressure and temperature. The strength of these preferences, however, decreases clearly with increasing temperature, as the height of these peaks are smaller at higher temperatures. This weakening of the orientational preferences leads to the complete washing out of the peaks in regions A and B of the surface at 575 K, indicating the complete loss of the orientational preferences of the water molecules in this case. The effect of the pressure on the orientational preferences is clearly much less pronounced than that of the temperature, as the orientational maps obtained at various pressures along the T ) 450 K isotherm look rather similar to each other. Nevertheless, it is also clear that the increase of the pressure leads to a small strengthening of the orientational preferences. Finally, it should be noted that the observed influence of the temperature and pressure on the orientational preferences of the surface water molecules is in qualitative agreement with what was found earlier by performing nonintrinsic analyses of the water surface.39,40 3.5. Lateral Interactions and Percolation at the Water Surface. At the boundary of an aqueous and an apolar phase the water molecules experience a rather unusual, asymmetric local environment, as they can form hydrogen bonds with other water molecules only in the direction of their own phase, whereas from the direction of the apolar phase they only experience a weak, van der Waals type attraction. This asymmetry of the local environment dictates the orientational preferences of the surface water molecules, as discussed in detail in the previous subsection. It is also known that, similarly to waters hydrating apolar solutes,84 the surface waters being in contact with a macroscopic apolar phase form, on average, stronger hydrogen bonds with each other than water molecules inside the bulk phase85 and that the lateral connectivity of the water molecules in the surface layer of their phase is considerably more extended than that in the consecutive subsurface molecular layers.56 This strong connectivity is also an important factor in determining the surface tension of water. Studying the connectivity of water molecules that are in contact with an apolar object is of great importance in understanding the general phenomenon of hydrophobic hydration and also in understanding particular but important problems related to the hydration of various hydrophobic objects of colloidal size, such as droplets of microemulsions or certain parts of the surface of various protein molecules.86



The investigation of the percolation properties of the surface waters at fluid interfaces, contrary to that in the first hydration shell of various large hydrophobic objects, clearly requires the use of an intrinsic surface analyzing method, such as ITIM, as the accurate knowledge of the list of the truly interfacial molecules is an obvious prerequisite of any reliable study of this kind. We have already studied the connectivity of the surface water molecules at the water-vapor47 and various water-organic liquid-liquid interfaces49,52,56 under ambient thermodynamic conditions. The general conclusion of these studies was that water molecules form a strongly percolating infinite twodimensional network in the surface layer, whereas no such lateral network exists in any of the subsequent molecular layers due to the three-dimensional nature of the hydrogen bonding water network in the bulk phase.56 The present study provides, however, a unique opportunity to investigate the temperature and pressure dependence of this percolating lateral network of surface waters, and, in particular, to study the break up of this infinite network with increasing temperature or decreasing pressure. In this study, we define two water molecules as being hydrogen bonded to each other if the distance of their O atoms is smaller than 3.35 Å and, at the same time, their shortest intermolecular O-H distance is smaller than 2.45 Å. These limiting distance values correspond to the first minimum position of the O-O and O-H radial distribution functions, respectively, of SPC/E water at ambient conditions. Two water molecules belong to the same lateral cluster if they are connected by a chain of intact hydrogen bonds, all of which are formed by two water molecules belonging to the same subsurface molecular layer. The size n of such a lateral cluster is simply defined by the number of water molecules belonging to it. In distinguishing between the two cases when the surface water molecules form an infinite, percolating lateral network and when they only form isolated hydrogen-bonding lateral oligomers, the following method is used here. At the percolation threshold the size distribution of the water clusters, P(n), obeys a power law:

P(n) ∼ n-R

(2)

with the universal exponent R ) 2.05 in two-dimensional systems.87 Thus, in the case of percolation, the P(n) size distribution of the system exceeds the critical line of eq 2 up to n values comparable with the total number of the surface water molecules, whereas below the percolation threshold the P(n) distribution drops below the critical line at small cluster size values and remains consistently below that at larger n values. The obtained P(n) curves of the surface water layer are shown and compared with the critical line of eq 2 in Figure 11 at various temperatures along the 100, 1000, 2500, and 5000 bar isobars. For better visualization, these data are shown on logarithmic scale, which transforms the critical curve of eq 2 to a straight line. As is seen, at 100 bar the surface waters form an infinite percolating network at every thermodynamic state point considered; thus, the breaking up of their percolating lateral network occurs above 450 K. At 1000 bar, the percolation threshold is located between 450 and 575 K, and at 2500 bar, it is located around 575 K (as the corresponding cluster size distribution agrees well with the critical line in a rather broad range of cluster sizes), whereas at 5000 bar it falls between 575 and 650 K. Further, considering also the fact that percolation of the surface waters has also been observed in state points 1 and 2 (i.e., at T ) 300 K, p ) 1 bar and T ) 375 K, p ) 2 bar,

Figure 11. Size distribution of the two-dimensional lateral hydrogenbonding clusters of the water molecules in the surface layer as obtained at various temperatures along the 5000 bar (top panel), 2500 bar (second panel), 1000 bar (third panel), and 100 bar (bottom panel) isobars. The data corresponding to the temperatures of 300, 375, 450, 575, and 650 K are shown by squares, circles, up triangles, down triangles, and diamonds, respectively. The critical line of the percolation threshold (see eq 2) is shown by a thick solid line. For better visualization, the data are shown on logarithmic scale.

respectively, data not shown), the percolation threshold is found to be located above 5000 bar along the 650 K, at about 2500 bar along the 575 K, and below 100, 2, and 1 bar along the 450, 375, and 300 K isotherms, respectively. The range within which the line of percolation is located is also shown in the phase diagram of the system (Figure 1) as estimated from these data. The obtained results reveal that the line of percolation of the surface waters clearly, by 200-400 K, precedes the upper critical line of the system in the phase diagram, indicating that the formation of a lateral percolating network of the surface water molecules at interfaces with an apolar fluid phase is by no means a universal behavior; instead, the existence of such a lateral network is also subject to the thermodynamic conditions. 4. Summary and Conclusions In this paper, we have presented a detailed analysis of the water-benzene interface in a broad range of thermodynamic states by means of Monte Carlo computer simulation and the novel ITIM method. A possible way of taking the mutual solubilities of the components in each other into account in an ITIM analysis has been proposed. Thus, the largest cluster of a given component (based on a sufficient and physically meaningful connectivity criterion) is regarded as the bulk phase of this component, whereas the other molecules of this type, i.e., the ones that do not belong to this largest cluster, are regarded as being dissolved in the other phase. Particular attention has been paid to the temperature and pressure dependence of the various characteristics of the interface. The obtained results not only stressed the need of using an intrinsic method, such as ITIM, in analyzing fluid interfaces, but also demonstrated how the systematic error

21692


originating from the incorrect identification of the interfacial and noninterfacial molecules, occurring when the interface is defined in a simple, nonintrinsic way through the average density profiles of the components, leads to a systematic error in the calculated thermodynamic properties (e.g., temperature of mixing) of the system. The obtained results showed that the reciprocal quantities related to the amplitude of the capillary waves corrugating the interface, such as the amplitude parameter a of the surface roughness or the mean width δ of the surface layer of both phases, show a linear dependence on the temperature, and they reach the value of zero (i.e., the corresponding original parameters themselves diverge) at the temperature of mixing of the two phases. Although the reciprocal mean width of the interface, D-1, has also been found to decrease linearly with the temperature, this quantity reaches zero value far above the mixing temperature. These results suggest that the temperatureinduced mixing of the two phases occurs with a mechanism in which both fluid surfaces become infinitely rough at the point of mixing, rather than the interfacial layer characterized by intermediate composition between the two phases becoming infinitely broad. We have also found a linear dependence between the reciprocal quantities characterizing the roughness and broadness of the surface layers and the logarithm of the pressure, but only above a certain temperature. Although it seems reasonable to assume that this temperature corresponds to the end point of the lower critical curve of the system, we could not test this hypothesis on the basis of the present calculations. We have found that the orientational preferences of the surface water molecules agree well with what was previously found at other water-apolar interfaces56 as well as at the surface of apolar solutes of varying size.83 These preferences are seen to be governed by the principle of maximizing the number of hydrogen-bonded neighbors of the surface water molecules. The observed orientational preferences themselves have been found to be insensitive to the thermodynamic conditions; however, their strength has been found to decrease with increasing temperature and decreasing pressure. Finally, the analysis of the lateral hydrogen-bonding network of the surface water molecules, previously found to span the entire water surface under ambient thermodynamic conditions at various water-apolar interfaces,56 has revealed that this spanning lateral network undergoes a percolation transition well, i.e., about 200-400 K below the mixing temperature of the system at every pressure. This result clearly indicates that the existence of such a percolating lateral hydrogen-bonding network of the surface water molecules is not a general feature of the water surface; instead, it depends also on the thermodynamic conditions. It should finally be noted that this behavior of the two-dimensional hydrogen-bonding network of the surface water molecules is in clear contrast with that of the threedimensional hydrogen-bonding network of the water molecules in the bulk phase, which was shown, both by experimental88,89 and computer simulation methods,90,91 to exist even above the critical point, and breaks down along the supercritical extension of the vapor-liquid coexistence line. Acknowledgment. This work has been supported by the Hungarian OTKA Foundation under Project No. OTKA 75328. References and Notes (1) Zhang, D.; Gutow, J. H.; Eisenthal, K. B.; Heinz, T. F. J. Chem. Phys. 1993, 98, 5099. (2) Huang, J. Y.; Wu, M. H. Phys. ReV. E 1994, 50, 3737.

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